Precalculus Examples

$\log (3 \sqrt{\frac{7}{q^{2 p}}})$

Step 1

Rewrite $\log (3 \sqrt{\frac{7}{q^{2 p}}})$ as $\log (3) + \log (\sqrt{\frac{7}{q^{2 p}}})$ .

$\log (3) + \log (\sqrt{\frac{7}{q^{2 p}}})$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{\frac{7}{q^{2 p}}}$ as ${(\frac{7}{q^{2 p}})}^{\frac{1}{2}}$ .

$\log (3) + \log ({(\frac{7}{q^{2 p}})}^{\frac{1}{2}})$

Step 3

Expand $\log ({(\frac{7}{q^{2 p}})}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\log (3) + \frac{1}{2} \log (\frac{7}{q^{2 p}})$

Step 4

Rewrite $\log (\frac{7}{q^{2 p}})$ as $\log (7) - \log (q^{2 p})$ .

$\log (3) + \frac{1}{2} (\log (7) - \log (q^{2 p}))$

Step 5

Expand $\log (q^{2 p})$ by moving $2 p$ outside the logarithm.

$\log (3) + \frac{1}{2} (\log (7) - (2 p \log (q)))$

Step 6

Multiply $2$ by $- 1$ .

$\log (3) + \frac{1}{2} (\log (7) - 2 (p \log (q)))$

Step 7

Remove parentheses.

$\log (3) + \frac{1}{2} (\log (7) - 2 p \log (q))$

Step 8

Simplify each term.

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Step 8.1

Apply the distributive property.

$\log (3) + \frac{1}{2} \log (7) + \frac{1}{2} (- 2 p \log (q))$

Step 8.2

Combine $\frac{1}{2}$ and $\log (7)$ .

$\log (3) + \frac{\log (7)}{2} + \frac{1}{2} (- 2 p \log (q))$

Step 8.3

Cancel the common factor of $2$ .

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Step 8.3.1

Factor $2$ out of $- 2 p \log (q)$ .

$\log (3) + \frac{\log (7)}{2} + \frac{1}{2} (2 (- p \log (q)))$

Step 8.3.2

Cancel the common factor.

$\log (3) + \frac{\log (7)}{2} + \frac{1}{2} (2 (- p \log (q)))$

Step 8.3.3

Rewrite the expression.

$\log (3) + \frac{\log (7)}{2} - p \log (q)$